Non-Grassmann mechanical model of the Dirac equation

Abstract

We construct a new example of the spinning-particle model without Grassmann variables. The spin degrees of freedom are described on the base of an inner anti-de Sitter space. This produces both ${\Gamma }^{\mu }$ and ${\Gamma }^{\mu \nu }$\,-matrices in the course of quantization. Canonical quantization of the model implies the Dirac equation. We present the detailed analysis of both the Lagrangian and the Hamiltonian formulations of the model and obtain the general solution to the classical equations of motion. Comparing {\it Zitterbewegung} of the spatial coordinate with the evolution of spin, we ask on the possibility of space-time interpretation for the inner space of spin. We enumerate similarities between our analogous model of the Dirac equation and the two-body system subject to confining potential which admits only the elliptic orbits of the order of de Broglie wave-length. The Dirac equation dictates the perpendicularity of the elliptic orbits to the direction of center-of-mass motion.

Latex, 48 pages, published version

References in corpus (17)

• Poincare covariant mechanics on noncommutative space
• Noncommutative relativistic particle on the electromagnetic background
• Quantum mechanics on noncommutative plane and sphere from constrained systems
• Conversion of second class constraints by deformation of Lagrangian local symmetries
• Quantization of Point-Like Particles and Consistent Relativistic Quantum Mechanics
• Classical mechanics of many particles defined on canonically deformed nonrelativistic space-time
• Local symmetries in the Hamiltonian framework. 1. Hamiltonian form of the symmetries and the Noether identities
• Classical Mechanics on Noncommutative Space with Lie-algebraic Structure
• Classical mechanics in reparametrization-invariant formulation and the Schrödinger equation
• Spinning-particle model for the Dirac equation and the relativistic Zitterbewegung
• Semiclassical Description of Relativistic Spin without use of Grassmann variables and the Dirac equation
• Classical description of spinning degrees of freedom of relativistic particles by means of commuting spinors
• Noncommutative Particles in Curved Spaces
• On the dynamics of classical particle with spin
• Reconstruction of higher stage first class constraints into the secondary ones
• Generalization of the Extended Lagrangian Formalism on a Field Theory and Applications
• Five-Dimensional Mechanics as the Starting Point for the Magueijo-Smolin Doubly Special Relativity

Cited by in corpus (4)

• Rigid particle revisited: extrinsic curvature yields the Dirac equation
• Geometric Constructions Underlying Relativistic Description of Spin on the Base of Non-Grassmann Vector-Like Variable
• Gravitating lepton bag model
• Classical gauge principle -- From field theories to classical mechanics